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Laplacian Eigenmodes for the Three Sphere

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Laplacian Eigenmodes for the Three Sphere Laplacian eigenmodes for the three sphere M. Lachi`eze-Rey Service d’Astrophysique, C. E. Saclay 91191 Gif sur Yvette cedex, France October 10, 2018 Abstract The vector space k of the eigenfunctions of the Laplacian on the 3 V three sphere S , corresponding to the same eigenvalue λk = k (k + 2 − k 2), has dimension (k + 1) . After recalling the standard bases for , V we introduce a new basis B3, constructed from the reductions to S3 of peculiar homogeneous harmonic polynomia involving null vectors. We give the transformation laws between this basis and the usual hyper-spherical harmonics. Thanks to the quaternionic representations of S3 and SO(4), we are able to write explicitely the transformation properties of B3, and thus of any eigenmode, under an arbitrary rotation of SO(4). This offers the possibility to select those functions of k which remain invariant under V a chosen rotation of SO(4). When the rotation is an holonomy transfor- mation of a spherical space S3/Γ, this gives a method to calculates the eigenmodes of S3/Γ, which remains an open probleme in general. We illustrate our method by (re-)deriving the eigenmodes of lens and prism space. In a forthcoming paper, we present the derivation for dodecahedral space. 1 Introduction 3 The eigenvalues of the Laplacian ∆ of S are of the form λk = k (k + 2), + − k where k IN . For a given value of k, they span the eigenspace of ∈ 2 2 V dimension (k +1) . This vector space constitutes the (k +1) dimensional irreductible representation of SO(4), the isometry group of S3. There are two commonly used bases (hereafter B1 and B2) for k V which generalize in some sense (see below) the usual spherical harmonics Yℓm for the two-sphere. The functions of these bases have a friendly be- arXiv:math/0401153v1 [math.SP] 14 Jan 2004 havior under some of the rotations of SO(4); this generalizes the property 3 of the Yℓm to be eigenfunctions of the angular momentum operator in IR . However, these functions show no peculiar properties under the general rotation of SO(4). Excepted for some cases (lens and prism spaces, see below), the search for the eigenmodes of the spherical spaces of the form S3/Γ remains an open problem. Since those are eigenmodes of S3 which remain invariant under the rotations of Γ, it is clear that this search requires an under- standing of the rotation properties of the basis functions under SO(4). The task of this paper is to examine the rotation properties of the eigenfunctions of k, as a preparatory work for the search for eigenfunc- V 1 tions of S3/Γ (in particular for dodecahedral space). This will be done through the introduction of a new basis B3 of k (in the case k even), V for which the rotation properties can be explicitely calculated: following a new procedure (that was already applied to S2 in [5]) we generate a system of coherent states on k. We extract from it a basis B3 of k, V V which seems to have been ignored in the literature, and presents original k properties. Each function ΦIJ of this basis B3 is defined as [the reduction to S3 of] an homogeneous harmonic polynomial in IR4, which takes the very simple form (X N)k. Here, the dot product extends the Euclidean · [scalar] dot product of IR4 to its complexification C4, and N is a null vector of C4, that we specify below. After defining these functions, we show that they form a basis of k, and we give the explicit transformation formulae V between B2 and B3. The properties of the basis B3 differ from those of the two other bases, and make it more convenient for particular applications. In particular, it is possible to calculate explicitely its rotation properties, under an arbi- trary rotation of SO(4), by using their quaternionic representation (section 3). This allows to find those functions which remain invariant under an arbitrary rotation. In section 4, we apply these result to (re-)derive the eigenmodes of lens and prism space. 2 Harmonic functions A function f on S3 is an eigenmode [of the Laplacian] if it verifies ∆f = + λf. It is known that eigenvalues are of the form λk = k (k +2), k IN . − ∈ k The corresponding eigenfunctions generate the eigen[vector]space , of V dimension (k + 1)2, which realizes an irreducible unitary representation of the group SO(4). First basis I call B1 the most widely used basis for k provided by the hyper- V spherical harmonics B1 ( kℓm Yℓm), ℓ = 1..k, m = ℓ..ℓ. (1) ≡ Y ∝ − It generalizes the usual spherical harmonics Yℓm on the sphere. In fact, it can be shown ([1], [2] p.240,[3]) that a basis of this type exists on any sphere Sn. Moreover, [2] [3] show that the B1 basis for Sn is “ naturally generated ” by the B1 basis for Sn−1. In this sense, the B1 basis for S3 2 is generated by the usual spherical harmonics Yℓm on the 2-sphere S . The generation process involves harmonic polynomials constructed from null complex vectors (see below). The basis B1 is in fact based on the reduction of the representation of SO(4) to representations of SO(3): each kℓm is an eigenfunction of an SO(3) subgroup of SO(4) which leaves Y a selected point of S3 invariant. This make these functions useful when one considers the action of that peculiar SO(3) subgroup. But they show no simple behaviour under a general rotation. We will no more use this basis. Second basis By group theoretical arguments, [1] construct a different ON basis of V k, which is specific to S3: B2 (Tk;m ,m ), m1,m2 = k/2...k/2, (2) ≡ 1 2 − where m1 and m2 vary independently by entire increments (and, thus, take entire or semi-entire values according to the parity of k). In the 2 spirit of the construction refered above, B2 may be seen as generated from a different choice of spherical harmonics on S2. The bases B1 and B2 appear respectively adapted to the systems of hyperspherical and toroidal (see below) coordinates to describe S3. The formula (27) of [1], reduced to the three-sphere, shows that the elements of this basis take a very convenient form if we use toroidal coordi- nates (as they are called by [7]) on the three sphere S3: (χ,θ,φ) spanning the range 0 χ π/2, 0 θ 2π and 0 φ 2π. They are conve- ≤ ≤ ≤ ≤ ≤ ≤ niently defined (see [7] for a more complete description) from an isometric embedding of S3 in IR4 (as the hypersurface x IR4; x = 1): ∈ | | x0 = r cos χ cos θ x1 = r sin χ cos φ x2 = r sin χ sin φ x3 = r cos χ sin θ µ 4 where (x ), µ = 0, 1,2, 3, is a point of IR . As shown in [7]), these coordinates appear naturally associated to some isometries. Very simple manipulations show that, with these coordinates, each eigenfunction of B2 takes the form: iℓθ imφ Tk;m ,m (X) tk;m ,m (χ) e e , (3) 1 2 ≡ 1 2 where the tk;m1,m2 (χ) are polynomials in cos χ and sin χ and we wrote, for simplification, ℓ m1 + m2, m m2 m1. ≡ ≡ − To have a convenient expression, we report this formula in the har- monic equation expressed in coordinates χ,θ,φ. This leads to a second order differential equation (cf. equ 15 of [7]). The solution is proportional ℓ m m,ℓ to a Jacobi polynomial: tk;m ,m (χ) cos χ sin χ P (cos 2χ), d 1 2 ∝ d ≡ k/2 m2. Thus, we have the final expression for the basis B2 − iθ ℓ iφ m (m,ℓ) Tk;m1,m2 (X)= Ck;m1,m2 [cos χ e ] [sin χ e ] Pd [cos(2χ)], (4) √(k+1) (k/2+m2)! (k/2−m2)! with Ck;m ,m − from normalization re- 1 2 ≡ π (k/2+m1)! (k/2 m1)! quirements (the variation rangesq of m1 anf m2 imply that the quantities under factorial sign are entire and positive). Note also the useful propor- tionality relations: ℓ m (m,ℓ) ℓ −m (−m,ℓ) cos χ sin χ P k−ℓ−m) (cos 2χ) cos χ sin χ P k−ℓ+m) (cos 2χ) 2 ∝ 2 −ℓ m (m,−ℓ) cos χ sin χ P k+ℓ−m (cos 2χ). ∝ 2 The term ζm ξn eiℓθ eimφ in (4) defines the rotation properties of ≡ Tk;m1,m2 under a specific subgroup of SO(4). This properties generalizes the properties of the spherical harmonics on the two-sphere S2, to be eigenfunctions of the rotation operator Px. This advantage has been used by [7] to calculate (from a slightly different basis) the eigenmodes of lens or prism spaces (see below section 4). However, the Tk;m1,m2 have no simple rotation properties under the general rotation of SO(4). This motivates the search for a different basis of k. V Note that the basis functions Tk;m1,m2 have also been introduced in [2] (p. 253), with their expression in Jacobi Polynomials. Note also that they are the complex counterparts of those proposed by [7] (their equ. 19) to find the eigenmodes of lens and prism spaces. The variation range of the indices m1,m2 here (equ. 2) is equivalent to their condition ℓ + m k, ℓ + m = k, mod (2), (5) | | | |≤ through the correspondence ℓ = m1 + m2, m = m2 m1.
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